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Polytope of Type {3,2,4,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,4,18}*864c
if this polytope has a name.
Group : SmallGroup(864,3999)
Rank : 5
Schlafli Type : {3,2,4,18}
Number of vertices, edges, etc : 3, 3, 4, 36, 18
Order of s0s1s2s3s4 : 9
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,2,4,18,2} of size 1728
Vertex Figure Of :
   {2,3,2,4,18} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {3,2,4,9}*432
   3-fold quotients : {3,2,4,6}*288b
   6-fold quotients : {3,2,4,3}*144
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,2,4,18}*1728, {6,2,4,18}*1728c
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 4,41)( 5,40)( 6,43)( 7,42)( 8,45)( 9,44)(10,47)(11,46)(12,49)(13,48)
(14,51)(15,50)(16,53)(17,52)(18,55)(19,54)(20,57)(21,56)(22,59)(23,58)(24,61)
(25,60)(26,63)(27,62)(28,65)(29,64)(30,67)(31,66)(32,69)(33,68)(34,71)(35,70)
(36,73)(37,72)(38,75)(39,74);;
s3 := ( 5, 6)( 8,12)( 9,14)(10,13)(11,15)(16,32)(17,34)(18,33)(19,35)(20,28)
(21,30)(22,29)(23,31)(24,36)(25,38)(26,37)(27,39)(41,42)(44,48)(45,50)(46,49)
(47,51)(52,68)(53,70)(54,69)(55,71)(56,64)(57,66)(58,65)(59,67)(60,72)(61,74)
(62,73)(63,75);;
s4 := ( 4,52)( 5,53)( 6,55)( 7,54)( 8,60)( 9,61)(10,63)(11,62)(12,56)(13,57)
(14,59)(15,58)(16,40)(17,41)(18,43)(19,42)(20,48)(21,49)(22,51)(23,50)(24,44)
(25,45)(26,47)(27,46)(28,68)(29,69)(30,71)(31,70)(32,64)(33,65)(34,67)(35,66)
(36,72)(37,73)(38,75)(39,74);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3, 
s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s2*s4*s3*s4*s2*s3*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(75)!(2,3);
s1 := Sym(75)!(1,2);
s2 := Sym(75)!( 4,41)( 5,40)( 6,43)( 7,42)( 8,45)( 9,44)(10,47)(11,46)(12,49)
(13,48)(14,51)(15,50)(16,53)(17,52)(18,55)(19,54)(20,57)(21,56)(22,59)(23,58)
(24,61)(25,60)(26,63)(27,62)(28,65)(29,64)(30,67)(31,66)(32,69)(33,68)(34,71)
(35,70)(36,73)(37,72)(38,75)(39,74);
s3 := Sym(75)!( 5, 6)( 8,12)( 9,14)(10,13)(11,15)(16,32)(17,34)(18,33)(19,35)
(20,28)(21,30)(22,29)(23,31)(24,36)(25,38)(26,37)(27,39)(41,42)(44,48)(45,50)
(46,49)(47,51)(52,68)(53,70)(54,69)(55,71)(56,64)(57,66)(58,65)(59,67)(60,72)
(61,74)(62,73)(63,75);
s4 := Sym(75)!( 4,52)( 5,53)( 6,55)( 7,54)( 8,60)( 9,61)(10,63)(11,62)(12,56)
(13,57)(14,59)(15,58)(16,40)(17,41)(18,43)(19,42)(20,48)(21,49)(22,51)(23,50)
(24,44)(25,45)(26,47)(27,46)(28,68)(29,69)(30,71)(31,70)(32,64)(33,65)(34,67)
(35,66)(36,72)(37,73)(38,75)(39,74);
poly := sub<Sym(75)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s4*s3*s2*s3*s4*s3*s4*s3, 
s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s2*s4*s3*s4*s2*s3*s2 >; 
 

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